Abstract
In this article, we present a technique that allows us to generate parallel tiled code to calculate general linear recursion equations (GLRE). That code deals with multidimensional data and it is computing-intensive. We demonstrate that data dependencies available in an original code computing GLREs do not allow us to generate any parallel code because there is only one solution to the time partition constraints built for that program. We show how to transform the original code to another one that exposes dependencies such that there are two linear distinct solutions to the time partition restrictions derived from these dependencies. This allows us to generate parallel 2D tiled code computing GLREs. The wavefront technique is used to achieve parallelism, and the generated code conforms to the OpenMP C/C++ standard. The experiments that we conducted with the resulting parallel 2D tiled code show that this code is much more efficient than the original serial code computing GLREs. Code performance improvement is achieved by allowing parallelism and better locality of the target code.
Highlights
Of each simpler relation is the left tuple of this relation, for example, for the first simpler relation, the left tuple is presented with variables (l, i, k ); the tuple after the sign → of each simpler relation is the right tuple of this relation, for example, for the first simpler relation, the right tuple is presented with variables (l 0, i, k0 ); the expressions after the sign | are the constraints of each simpler relation, each constraint is represented with the conjunctions of inequalities built on tuple variables and parameters; and ∧ is the logical AND operator
We presented an approach to generate parallel tiled code for computing general linear recurrence equations (GLREs) presented in Listing 1
We demonstrated how to transform that code to obtain the modified code shown in Listing 3, which exposes dependencies such that there exist two linear independent solutions to the time partition constraints formed on the basis of those dependencies
Summary
The purpose of this document is to show a way to produce a parallel program for computing general linear recurrence equations (GLREs). All known techniques based on affine transformations and/or transitive closure of dependence graphs [1,2] are unable to generate any tiled code (serial and/or parallel) for the code in Listing 1. Those techniques are within the class of reordering transformations. They do not introduce any additional computations to generated target code in comparison with an original code; they only reorder iterations of the original code allowing for tiling and/or parallelism
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